Estimates and monotonicity for a heat flow of isometric G2-structures

Abstract

Given a $7$-dimensional compact Riemannian manifold $\left(M,g\right)$ that admits $G_2$-structure, all the $G_2$-structures that are compatible with the metric $g$ are parametrized by unit sections of an octonion bundle over $M$. We define a natural energy functional on unit octonion sections and consider its associated heat flow. The critical points of this functional and flow precisely correspond to $G_2$-structures with divergence-free torsion. In this paper, we first derive estimates for derivatives of $V\left(t\right)$ along the flow and prove that the flow exists as long as the torsion remains bounded. We also prove a monotonicity formula and and an $\epsilon$-regularity result for this flow. Finally, we show that within a metric class of $G_2$-structures that contains a torsion-free $G_2$-structure, under certain conditions, the flow will converge to a torsion-free $G_2$-structure.

43 pages. Version 3: fixed typos, and minor updates for clarity. Added journal info. Version 2: This version acknowledges a preprint by Dwivedi, Gianniotis, and Karigiannis (arXiv:1904.10068) that was posted a couple of days after Version 1 of this preprint had been posted, and has a substantial but independent overlap with this preprint. Also, an error in Corollary 7.2 is corrected

References in corpus (1)

• Local geometry of the G2 moduli space

Cited by in corpus (8)

• Harmonic flow of geometric structures
• A gradient flow of isometric ${\mathrm{G}}_2$ structures
• Flows of ${\mathrm{G}}_2$-structures on contact Calabi--Yau $7$-manifolds
• Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere
• Harmonic $G_2$-structures on almost Abelian Lie groups
• Some remarks on strong ${\mathrm{G}}_2$-structures with torsion
• The Space of Closed $G_2$-Structures. I. Connections
• Cohomogeneity one solitons for the isometric flow of $G_2$-structures